Question

solve the system by graphing. if the system does not have the unique solution, also state the number of solutions and whether the system is dependent or inconsistent.
2x + 3y=-2
x - 4y=-1
part 0 / 2
part 1 of 2
graph the lines.

Explanation:

Step1: Rewrite equations in slope - intercept form

For the equation \(2x + 3y=-2\), solve for \(y\):
\[

$$\begin{align*} 3y&=-2x - 2\\ y&=-\frac{2}{3}x-\frac{2}{3} \end{align*}$$

\]
For the equation \(x - 4y=-1\), solve for \(y\):
\[

$$\begin{align*} -4y&=-x - 1\\ y&=\frac{1}{4}x+\frac{1}{4} \end{align*}$$

\]

Step2: Find the intersection point

Set the two equations equal to each other:
\[-\frac{2}{3}x-\frac{2}{3}=\frac{1}{4}x+\frac{1}{4}\]
Multiply through by 12 to clear the fractions:
\[

$$\begin{align*} 12\times(-\frac{2}{3}x)-12\times\frac{2}{3}&=12\times\frac{1}{4}x + 12\times\frac{1}{4}\\ -8x-8&=3x + 3 \end{align*}$$

\]
Add \(8x\) to both sides: \(-8 = 11x+3\)
Subtract 3 from both sides: \(-11 = 11x\)
Solve for \(x\): \(x=-1\)
Substitute \(x = - 1\) into \(y=\frac{1}{4}x+\frac{1}{4}\), we get \(y=\frac{1}{4}\times(-1)+\frac{1}{4}=0\)

Answer:

The solution of the system is \(x=-1,y = 0\)